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A097727 Pell equation solutions (5*b(n))^2 - 26*a(n)^2 = -1 with b(n)=A097726(n), n>=0. 8
1, 101, 10301, 1050601, 107151001, 10928351501, 1114584702101, 113676711262801, 11593909964103601, 1182465139627304501, 120599850332020955501, 12300002268726510156601, 1254479631559772015017801, 127944622416828019021659101, 13049097006884898168194210501 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Hypotenuses of primitive Pythagorean triples in A195622 and A195623. - Clark Kimberling, Sep 22 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Tanya Khovanova, Recursive Sequences

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (102, -1).

FORMULA

a(n) = S(n, 2*51) - S(n-1, 2*51) = T(2*n+1, sqrt(26))/sqrt(26), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.

a(n) = ((-1)^n)*S(2*n, 10*I) with the imaginary unit I and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

G.f.: (1-x)/(1-102*x+x^2).

a(n) = 102*a(n-1) - a(n-2) for n>1 ; a(0)=1, a(1)=101. - Philippe Deléham, Nov 18 2008

EXAMPLE

(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 =-1.

MATHEMATICA

LinearRecurrence[{102, -1}, {1, 101}, 20] (* Harvey P. Dale, Apr 12 2014 *)

CoefficientList[Series[(1-x)/(1-102x+x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Apr 13 2014 *)

PROG

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-102*x+x^2)) \\ G. C. Greubel, Aug 01 2019

(MAGMA) I:=[1, 101]; [n le 2 select I[n] else 102*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

(Sage) ((1-x)/(1-102*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

(GAP) a:=[1, 101];; for n in [3..20] do a[n]:=102*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

CROSSREFS

Cf. A097725 for S(n, 102).

Row 5 of array A188647.

Sequence in context: A071783 A082808 A100027 * A083981 A267779 A138148

Adjacent sequences:  A097724 A097725 A097726 * A097728 A097729 A097730

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

EXTENSIONS

More terms from Harvey P. Dale, Apr 12 2014

STATUS

approved

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Last modified November 16 20:13 EST 2019. Contains 329206 sequences. (Running on oeis4.)